Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons

Abstract

Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, an observable that rapidly approaches a value proportional to the Zak phase during the free evolution of the system. Then we measure the Zak phase in a photonic quantum walk of twisted photons, by observing the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general and readily applicable to all present one-dimensional platforms simulating static or Floquet chiral systems.

Topological characterization of chiral models through their long time dynamics

“…These states are not protected by the chiral symmetry, and therefore not robust against (chiral-preserving) disorder. In order to illustrate this fact, we add a spatial disorder in the operator W: the hoppings of the Hamiltonian of W are multiplied by a factor 1  + ( ), where ò is a random number in the rangeThe right side of the energy spectrum (after the dashed line) in figure 6(b) shows clearly that, whereas the 0 and the p-energy states remain unaffected, the unprotected states change of energy when disorder is applied.The effect of spatial disorder and noise on the mean chiral displacement for systems with 2  = were already discussed at length in our previous publication, [24]. In particular, there we confirmed that this observable is a robust topological marker by showing that, in presence of chiral-preserving static spatial disorder of amplitude small compared to the gap, the ensemble average of the mean chiral displacement smoothly converges to the value obtained for a clean system.…”

“…The second one is to measure the response to an external change. The observation of edge states [11,21,30,31] and the measurement of the winding number through the mean chiral displacement [24] belong to the first category. The measurement of the winding number by interferometric architectures [6,25], by introducing losses [15,32,33], and by scattering measurements [22] belong to the second category.…”

“…The measurement of the winding number by interferometric architectures [6,25], by introducing losses [15,32,33], and by scattering measurements [22] belong to the second category. Here we generalize the notion of mean chiral displacement introduced in [24], and we present an intrinsic measurement of the topology for 1D chiral Hamiltonians with an arbitrary (even) number of sites per unit cell. This measure is based on the real-time evolution of an initially localized, single-particle state.…”

“…The topological invariant of the bulk, the winding number  , allows one to predict the number of zero energy edge states. 1D chiral topological insulators have been realized in numerous platforms as ultracold atoms [6,11], photonic crystals [15], photonic quantum walks [21][22][23][24][25]. Let us notice that the 1D chiral Hamiltonian can be static or the effective Hamiltonian of a Floquet system.…”

“…At a qualitative level, systems with internal dimension 2  > behave in no manner differently from systems with 2  = in presence of disorder. Readers interested in this topic are therefore referred to our earlier publication [24], and to its supplemental information, where the matter is discussed in great detail. …”